cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A375361 Odd numbers with at least two prime divisors of the form 4*k + 1 counted with multiplicity.

Original entry on oeis.org

25, 65, 75, 85, 125, 145, 169, 175, 185, 195, 205, 221, 225, 255, 265, 275, 289, 305, 325, 365, 375, 377, 425, 435, 445, 455, 475, 481, 485, 493, 505, 507, 525, 533, 545, 555, 565, 575, 585, 595, 615, 625, 629, 663, 675, 685, 689, 697, 715, 725, 745, 765, 775
Offset: 1

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Author

Arkadiusz Wesolowski, Aug 13 2024

Keywords

Comments

Odd numbers k such that k^2 can be expressed as the arithmetic mean of two distinct perfect squares in more than one way. For example, 25^2 = (5^2 + 35^2)/2 = (17^2 + 31^2)/2.
Let x be a squared integer which is the central element of a 3 X 3 magic square in which seven (or more) of the entries are squared integers. If the greatest common divisor of all nine entries is 1, then the square root of x is a composite number that is divisible only by primes congruent to 1 mod 4. For example, sqrt(A221669(5)) = 425 is both in A004613 and in this sequence.

Examples

			65 is in this sequence because 65 has two prime factors of the form 4*k + 1, namely 5 = 4*1 + 1 and 13 = 4*3 + 1.

Crossrefs

Cf. A004613, A008844, A221669, A221670, A268855, A309812, A375459.

Programs

  • Magma
    f:=func; nopf:=func; sum:=func; [n: n in [1..775 by 2] | sum(n) gt 1];
  • PARI
    isok(n) = my(v=Vec(factor(n))); n%2&&sum(t=1, omega(n), if((v[1]%4)[t]==1, v[2][t]))>1;
  • PARI
    isok(n) = my(t); if(n%2, for(k=sqrtint(n^2-1)+2, sqrtint(2*n^2-1), if(issquare(2*n^2-k^2)&&t++>1, return(1)))); 0;

A325421 Numbers k coprime to 10 such that there are exactly two values of A for which k^2+4*A and k^2-4*A are perfect squares.

Original entry on oeis.org

169, 289, 507, 841, 867, 1183, 1369, 1521, 1681, 1859, 2023, 2523, 2601, 2809, 3179, 3211, 3549, 3721, 3887, 4107, 4563, 5043, 5239, 5329, 5491, 5577, 5887, 6069, 6647, 7267, 7569, 7803, 7921, 7943, 8281, 8427, 8959, 9251, 9409, 9537, 9583, 9633, 9971
Offset: 1

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Author

Mohsin A. Shaikh, Sep 06 2019

Keywords

Comments

These are the odd integers k, not a multiple of 5, such that k^2 is an arithmetic mean of two other odd perfect squares in exactly two ways.

Examples

			169 is a term since 169^2±4*(5070) and 169^2±4*(7140) are all perfect squares.

Crossrefs

Cf. A002144, A309812.

Programs

  • PARI
    ok(k)={if(k%2==0||k%5==0, 0, my(k2=k^2, L=List()); forstep(i=1, k-1, 2, my(d=k2-i^2); if(issquare(k2+d), listput(L,i))); #L==2)}
for(k=1, 10000, if(ok(k), print1(k, ", "))) \\ Andrew Howroyd, Sep 06 2019

Extensions

a(28)-a(43) from Andrew Howroyd, Sep 06 2019

A382117 a(n) = sum (-1)^(((x - 1)*(y - 1))/4), where x and y are coprime positive integers, equidistant from n, such that x <= y.

Original entry on oeis.org

1, 1, 1, 2, 0, 2, 1, 4, 1, 4, 1, 4, 0, 6, 0, 8, 0, 6, 1, 8, 2, 10, 1, 8, 0, 12, 1, 12, 0, 8, 1, 16, 2, 16, 0, 12, 0, 18, 0, 16, 0, 12, 1, 20, 0, 22, 1, 16, 1, 20, 0, 24, 0, 18, 0, 24, 2, 28, 1, 16, 0, 30, 2, 32, 0, 20, 1, 32, 2, 24, 1, 24, 0, 36, 0, 36, 2, 24
Offset: 1

Views

Author

Mike Jones, Apr 26 2025

Keywords

Comments

Sequence inspired by the law of quadratic reciprocity.

Examples

			a(4)=2 because (-1)^(((1 - 1)*(7 - 1))/4) + (-1)^(((3 - 1)*(5 - 1))/4) = (-1)^0 + (-1)^2 = 1 + 1 = 2.

Crossrefs

Cf. A309812 (indices of zeros).

Programs

  • PARI
    a(n) = sum(x=1, n, my(y=2*n-x); if ((x<=y) && (gcd(x, y)==1), (-1)^(((x-1)*(y-1))/4))); \\ Michel Marcus, Apr 27 2025

Extensions

More terms from Alois P. Heinz, Apr 26 2025
Showing 1-3 of 3 results.

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